- Open Access
Positive periodic solutions of second-order difference equations with weak singularities
© Ma and Lu; licensee Springer. 2012
- Received: 28 March 2012
- Accepted: 27 June 2012
- Published: 27 June 2012
We study the existence of positive periodic solutions of the second-order difference equation
via Schauder's fixed point theorem, where a, c : ℤ → ℝ+ are T -periodic functions, f ∈ C(ℤ × (0, ∞), ℝ) is T -periodic with respect to t and singular at u = 0.
Mathematics Subject Classifications: 34B15.
- positive periodic solutions
- difference equations
- Schauder's fixed point theorem
- weak singularities.
where a, c : ℤ → ℝ+ are T-periodic functions, f ∈ C(ℤ × (0, ∞), ℝ) is T-periodic with respect to t and singular at u = 0.
where a ∈ L1(ℝ/T ℤ; ℝ+), c ∈ L1(R/T ℤ; ℝ), f ∈ Car(ℝ/T ℤ × (0, ∞), ℝ) and is singular at u = 0, see [7–20]. The first existence result with weak force condition appears in Rachunková et al. . Since then, the Equation (1.3) with f has weak singularities has been studied by several authors, see Torres [17, 18], Franco and Webb , Chu and Li .
Recently, Torres  showed how a weak singularity can play an important role if Schauder's fixed point theorem is chosen in the proof of the existence of positive periodic solution for (1.3). For convenience, for a given function ξ ∈ L∞[0, T], we denote the essential supremum and infimum of ξ by ξ* and ξ*, respectively. We write ξ ≻ 0 if ξ ≥ 0 for a.e. t ∈ [0, T] and it is positive in a set of positive measure. Under the assumption
Torres showed the following three results
If γ * > 0, then there exists a positive T-periodic solution of (1.3).
Theorem B. [, Theorem 2] Let (H1) hold. Assume that
If γ* = 0. Then (1.3) has a positive T-periodic solution.
Then (1.3) has a positive T-periodic solution.
However, the discrete analogue of (1.3) has received almost no attention. In this article, we will discuss in detail the singular discrete problem (1.1) with our goal being to fill the above stated gap in the literature. For other results on the existence of positive solution for the other singular discrete boundary value problem, see [21–24] and their references. From now on, for a given function ξ ∈ l∞(0, ∞), we denote the essential supremum and infimum of ξ by ξ* and ξ*, respectively. We write ξ ≻ 0 if ξ ≥ 0 for t [0, T ]ℤ and it is positive in a set of positive measure.
Our main results are the following
Theorem 1.1. Let (A1) and (A2) hold. If γ* > 0. Then (1.1) has a positive T-periodic solution.
Theorem 1.2. Let (A1) and (A3) hold. If γ* = 0. Then (1.1) has a positive T-periodic solution.
Then (1.1) has a positive T-periodic solution.
where ε, η > 0. Obviously, f0 satisfies (A2) with M = m = 1, b(t) = e(t) ≡ 1. However, it is fail to satisfy (H2) since it can not be bounded by a single function for any γ ∈ (0, ∞) and any h ≻ 0. □
Remark 1.2. If ε, η ∈ (0, 1), then the function f0 defined by (1.6) satisfies (A3) with ν = μ = η, α = β = ε, and b1(t) ≡ b2(t) ≡ e(t) ≡ 1. However, it is fail to satisfy (H3). □
under the norm. Then (X, || · ||) is a Banach space.
into itself, where R > r > 0 are positive constants to be fixed properly.
Therefore, A(K) ⊂ K if r = γ* and , and the proof is finished. □
By a direct application of Schauder's fixed point theorem, the proof is finished if we prove that A maps the closed convex set K into itself, where R and r are positive constants to be fixed properly and they should satisfy R > r > 0 and R > 1.
and these inequalities hold for R big enough because σ < 1 and ν < 1. □
Remark 3.1. It is worth remarking that Theorem 1.2 is also valid for the special case that c(t) ≡ 0, which implies that γ* = 0. □
By a direct application of Schauder's fixed point theorem, the proof is finished if we prove that A maps the closed convex set K into itself, where R and r are positive constants to be fixed properly and they should satisfy R > 1 > r > 0.
where J i (i = 1, 2) is defined as in Section 3 and .
This completes the proof. □
and is a constant.
Consequently, Theorem 1.3 yields that (4.2) has a positive solution. □
The authors are very grateful to the anonymous referees for their valuable suggestions. This work was supported by the NSFC (No. 11061030), NSFC (No. 11126296), the Fundamental Research Funds for the Gansu Universities.
- Atici FM, Guseinov GS: Positive periodic solutions for nonlinear difference equations with periodic coefficients. J Math Anal Appl 1999, 232: 166–182. 10.1006/jmaa.1998.6257MathSciNetView ArticleMATHGoogle Scholar
- Atici FM, Cabada A: Existence and uniqueness results for discrete second-order periodic boundary value problems. Comput Math Appl 2003, 45: 1417–1427. 10.1016/S0898-1221(03)00097-XMathSciNetView ArticleMATHGoogle Scholar
- Atici FM, Cabada A, Otero-Espinar V: Criteria for existence and nonexistence of positive solutions to a discrete periodic boundary value problem. J Diff Equ Appl 2003, 9(9):765–775. 10.1080/1023619021000053566MathSciNetView ArticleMATHGoogle Scholar
- Ma R, Ma H: Positive solutions for nonlinear discrete periodic boundary value problems. J Appl Math Comput 2010, 59: 136–141. 10.1016/j.camwa.2009.07.071View ArticleMathSciNetMATHGoogle Scholar
- He T, Xu Y: Positive solutions for nonlinear discrete second-order boundary value problems with parameter dependence. J Math Anal Appl 2011, 379(2):627–636. 10.1016/j.jmaa.2011.01.047MathSciNetView ArticleMATHGoogle Scholar
- Ma R, Lu Y, Chen T: Existence of one-signed solutions of discrete second-order periodic boundary value problems. Abstr Appl Anal 2012, 2012: 13. (Article ID 437912)MathSciNetMATHGoogle Scholar
- Lazer AC, Solimini S: On periodic solutions of nonlinear differential equations with singularities. Proc Am Math Soc 1987, 99: 109–114. 10.1090/S0002-9939-1987-0866438-7MathSciNetView ArticleMATHGoogle Scholar
- Gordon WB: Conservative dynamical systems involving strong forces. Trans Am Math Soc 1975, 204: 113–135.View ArticleMathSciNetMATHGoogle Scholar
- Gordon WB: A minimizing property of Keplerian orbits. Am J Math 1977, 99: 961–971. 10.2307/2373993View ArticleMathSciNetMATHGoogle Scholar
- Bonheure D, Fabry C, Smets D: Periodic solutions of forced isochronous oscillators at resonance. Discret Contin Dyn Syst 2002, 8(4):907–930.MathSciNetView ArticleMATHGoogle Scholar
- Fonda A, Mansevich R, Zanolin F: Subharmonics solutions for some second order differential equations with singularities. SIAM J Math Anal 1993, 24: 1294–1311. 10.1137/0524074MathSciNetView ArticleMATHGoogle Scholar
- Jiang D, Chu J, Zhang M: Multiplicity of positive periodic solutions to superlinear repulsive singular equations. J Diff Equ 2005, 211(2):282–302. 10.1016/j.jde.2004.10.031MathSciNetView ArticleMATHGoogle Scholar
- del Pino M, Manásevich R, Montero A: T -periodic solutions for some second order differential equations with singularities. Proc R Soc Edinburgh Sect A 1992, 120(3–4):231–243. 10.1017/S030821050003211XView ArticleMathSciNetMATHGoogle Scholar
- Rachunková I, Staněk S, Tvrdý M: Singularities and Laplacians in Boundary Value Problems for Nonlinear Ordinary Differential Equations, Handbook of Differential Equations (Ordinary Differential Equations). Elsevier, Amsterdam 2006., 3:Google Scholar
- Torres PJ, Zhang M: Twist periodic solutions of repulsive singular equations. Non-linear Anal 2004, 56: 591–599.MathSciNetView ArticleMATHGoogle Scholar
- Rachunková I, Tvrdý M, Vrkoč I: Existence of nonnegative and nonpositive solutions for second order periodic boundary value problems. J Diff Equ 2001, 176: 445–469. 10.1006/jdeq.2000.3995View ArticleMathSciNetMATHGoogle Scholar
- Torres PJ: Existence of one-signed periodic solutions of some second order differential equations via a Krasnoselskii fixed point theorem. J Diff Equ 2003, 190: 643–662. 10.1016/S0022-0396(02)00152-3View ArticleMathSciNetMATHGoogle Scholar
- Torres PJ: Weak singularities may help periodic solutions to exist. J Diff Equ 2007, 232: 277–284. 10.1016/j.jde.2006.08.006View ArticleMathSciNetMATHGoogle Scholar
- Franco D, Webb JKL: Collisionless orbits of singular and nonsingular dynamical systems. Discret Contin Dyn Syst 2006, 15: 747–757.MathSciNetView ArticleMATHGoogle Scholar
- Chu J, Li M: Positive periodic solutions of Hill's equations with singular nonlinear perturbations. Nonlinear Anal 2008, 69: 276–286. 10.1016/j.na.2007.05.016MathSciNetView ArticleMATHGoogle Scholar
- Agarwal RP, O'Regan D: Singular discrete boundary value problems. Appl Math Lett 1999, 12: 127–131.View ArticleMathSciNetMATHGoogle Scholar
- Agarwal RP, Perera K, O'Regan D: Multiple positive solutions of singular and nonsingular discrete problems via variational methods. Nonlinear Anal 2004, 58: 69–73. 10.1016/j.na.2003.11.012MathSciNetView ArticleMATHGoogle Scholar
- Lü H, O'Regan D, Agarwal RP: Positive solution for singular discrete boundary value problem with sign-changing nonlinearities. J Appl Math Stoch Anal 2006, 2006: 1–14. (Article ID 46287)View ArticleMATHGoogle Scholar
- Jiang D, Pang PYH, Agarwal RP: Upper and lower solutions method and a superlinear singular discrete boundary value problem. Dyn Syst Appl 2007, 16: 743–754.MathSciNetMATHGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.